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Stochastic Convenience Yield, OptimalHedging and the Term Structure of Open
Interest and Futures Prices
Harrison Hong ∗
July 23, 2001
Abstract
This paper develops a dynamic, equilibrium model of a futures market to study optimalhedging and the term structure of open interest and futures prices. Investors continuouslyface spot price risk over time and attempt to hedge this risk using futures. Convenienceyield shocks generate basis risk to rolling over near-to-maturity futures. Hence, investorsneed to simultaneously trade far-from-maturity futures. The model predicts that in mar-kets with substantial and mean reverting convenience yield shocks (e.g. energy futures),open interest is evenly distributed among contracts of different maturities. In marketswhere these shocks are persistent (e.g. metal futures), open interest is concentrated innear-to-maturity futures. The model generates additional implications regarding how theterm structure of futures price volatility and the futures risk premium depend on thenature of convenience yield shocks.
JEL No(s): G11, G12, G13
Keywords: Hedging, Basis Risk, Open Interest, Futures Prices
∗Thanks to Darrell Duffie, Glenn Ellison, Jeremy Stein and Jiang Wang for helpful comments. Gradu-ate School of Business, Stanford University. Address correspondence to Harrison Hong, Graduate Schoolof Business, Stanford University, Stanford, CA 94305. Email: [email protected].
There is substantial variation across futures markets in the pattern of open interest with
respect to contract maturity. In futures markets such as S&P and currencies, most of
the open interest is concentrated in the near-to-maturity (or nearby) contracts, with the
front two contracts typically accounting for about 80% of the total open interest in these
markets. In markets such as metals and agriculturals, open interest tends to be less
concentrated in nearby contracts. And in oil and other energy markets, open interest is
evenly distributed between nearby and far-from-maturity (distant) futures.1
What accounts for this cross-sectional variation in the term structure of open interest?
This question is an important one for at least a couple of reasons. First, these equilibrium
open interest patterns reflect the variation of optimal hedging and speculation strategies
across markets. So, an answer to this question will enhance our understanding of the
determinants of optimal hedging and speculation in futures markets. Importantly, a
number of papers have documented that the net holdings of hedgers, after controlling
for systematic risk, significantly affect futures returns (see, e.g., Chang (1985), Carter,
Rausser and Schmitz (1983), Bessembinder (1992), de Roon, Nijman and Veld (2000)).
Hence an understanding of the determinants of the term structure of open interest will
enhance our understanding of the term structure of futures prices.
The analysis of the determinants of optimal hedging, speculation and futures prices
has a long and distinguished history. In the original normal backwardation theory of
Keynes and Hicks (see, e.g., Keynes (1923) and Hicks (1939)), producers short futures
to hedge their initially long positions in the underlying spot. Their supply of futures, or
hedging pressure, tends to drive down the futures price relative to the expected value of
the later spot price. Hence, risk averse speculators who long the futures to share the risk
are provided with a positive expected return.
A substantial literature has since generalized the theory of Keynes-Hicks in a number
of directions such as allowing for 1) producers to face quantity risks as well as price risks
(see, e.g., Rolfo (1980), Newberry and Stigliz (1981)); 2) multiple consumption goods (see,
e.g., Stiglitz (1983), Britto (1984)); and 3) a stock market or the tradeability of equity
claims to producers’ future revenues (see, e.g., Stoll (1979), Hirshleifer (1988, 1989)).
1These patterns are apparent from looking up futures prices and open interest in the Wall StreetJournal on any given day. Futures textbooks also discuss these patterns.
1
Papers in the literature have also addressed how uncertainty resolution affects optimal
dynamic hedging decisions (see, e.g., Anderson and Danthine (1983), Duffie and Jackson
(1990), Hirshleifer (1991)) and how asymmetric information among market participants
impacts equilibrium outcomes (see, e.g., Grossman (1977), Bray (1981), Hong (2000)).2
However, this literature is largely silent on why the term structure of open interest
varies across futures markets. Most of the models in this literature allow hedgers to
trade only one futures contract at any point in time. It is often assumed that market
participants want to hedge against spot price risk at a future date and that a futures
is available to hedge the risk on this particular date. So hedgers typically do not need
to simultaneously trade futures of differing maturities. Within this canonical framework,
differences in open interest patterns across futures markets need to be traced to differing
hedging horizons among market participants. It is not clear, however, if and why hedgers
in oil markets have a longer hedging horizon than hedgers in metals or currencies.
What does differ substantially across futures markets is the stochastic nature of conve-
nience yield. In S&P and currency futures, there is a negligible convenience yield. In agri-
cultural and metal markets, there is a more substantial convenience yield that fluctuates
over time. And in energy futures, especially oil, convenience yield fluctuations are notori-
ously large. Not only does the variance of convenience yield shocks vary across markets,
but so does the persistence of these shocks (see, e.g., Bessembinder e.t.al. (1995)). For
instance, convenience yield shocks in energy markets tend to be very transitory, whereas
such shocks in metal markets tend to be persistent.
The stochastic nature of convenience yield can have important effects on the term
structure of open interest. The following examples bring out the intuition for why this
is the case. In markets such as S&P futures, which are essentially cost-of-carry, even
if one wanted to hedge over a 2 year horizon, one can do a pretty good job by rolling
the 3-month contract. One does not really need a 2-year contract in the absence of any
convenience yield shocks because there is negligible basis risk. In contrast, oil futures of
differing maturities are not good substitutes because of the tremendous convenience yield
2Models in which futures prices depend only on the degree of covariation between futures prices andchanges in economic state variables (systematic risk) without a role for hedging pressure include Black(1976), Breeden (1980), and Richard and Sundaresan (1981). Evidence for these models is mixed at best(see, e.g., Dusak (1973), Hodrick and Srivastava (1983), Jagannathan (1985)).
2
fluctuations. Trying to hedge over a long horizon by rolling the short-term contract can
be quite costly because of basis risk, as the case of Metallgesellschaft (or MG Corp) makes
clear. In a sense, these two examples make clear that to optimally hedge spot positions,
it is more essential to trade a range of maturities in oil than in S&P.
In this paper, I develop a model of a futures market to explore these ideas. In my
model, holding a spot position (as opposed to futures positions) provides the investor
with an exogenously specified convenience yield, which is stochastic and mean reverting.
Market participants are infinitely lived and are not just exposed to spot price risk at a
particular future date. Instead, their spot positions are continually subject to price risk
over time. Investors can hedge their spot positions with two futures contracts (a nearby
and a distant). It is assumed that the available set of securities does not span fluctuations
in the convenience yield. Hence futures are non-redundant securities and cannot be priced
by arbitrage. Instead, prices are determined jointly by the hedging and speculative trades
of investors.
The term structure of open interest along with the resulting futures prices depend
on the properties of the convenience yield. Because of a fluctuating convenience yield,
positions in the nearby are subject to basis risk. Hence, investors optimally hedge their
positions in the nearby using the distant futures. This basis risk to trading in the nearby
depends on the persistence of convenience yield shocks. I show that when these shocks
are very persistent, there is less basis risk in rolling over the nearby. So rolling over the
nearby is a good substitute for trading in the distant contract, and the distant futures
atttracts less open interest. One testable implication then is that all else equal, markets
with more mean reverting convenience yield shocks should have open interest more evenly
distributed between nearby and distant contracts.
More importantly, since all prices in the economy are determined endogenously, the
model can speak to a number of features of the term structure of prices. For instance,
in the presence of mean-reverting convenience yield shocks, the price of a given futures
is less sensitive to contemporaneous shocks than the spot price. These mean-reverting
shocks are less reflected in the price of the futures than the spot because they will die away
by the time the futures expires. By the same logic, the distant contract is less sensitive
to underlying shocks than the nearby contract. So, the return of the nearby contract is
3
more volatile than that of the distant contract. This term structure of volatility pattern
is known as the “Samuelson effect” (see Samuelson (1965)). The Samuelson effect is more
pronounced in markets with more mean reverting spot price shocks.
Combining the results for the term structure of open interest and price volatility, my
model predicts that markets with mean reverting convenience yield shocks should have
open interest evenly distributed among contract maturities and should also exhibit a
pronounced Samuelson Effect. Furthermore, I also show that markets with more volatile
convenience yield shocks are more likely to have a larger futures risk premium. And the
futures risk premium increases with the persistence of convenience yield shocks.
My model is related to some previous work on the term structure of open interest
and futures prices.3 A number of other papers consider the behavior of optimal hedging
over time (see, e.g., Anderson and Danthine (1983), Kamara (1993), Hong (2000)). These
papers consider a futures contract expiring a number of periods in the future. They
analyze the behavior of optimal hedging and futures prices as the contract expires. They
do not generally allow trading in multiple futures contracts at the same time. In contrast,
investors need to simultaneously trade multiple futures contracts at the same time in my
model. This feature is crucial since the empirical findings regarding the term structure
of open interest and futures prices really speak to the simultaneous trading of futures of
different maturities at a point in time.
One paper that does allow for trading in multiple futures is Duffie and Jackson (1990),
who develop a model of optimal hedging and equilibrium in a dynamic futures market.
The intent of their paper is to solve for (in closed-form) optimal hedging formulas under a
variety of settings. Their model is more general than mine in some ways such as allowing
investors to trade more than two futures of different maturities at the same time. Unlike
my model, the spot price in their model is exogenously specified and the non-marketed
risk process that my investors face is different than theirs. As such, they do not consider
the effects of convenience yield fluctuations on optimal hedging and equilibrium and my
focus differs from theirs.
3My model is also related to a number of papers on optimal hedging of long-term commitmentswith short-term futures inspired in part by the MG Corp debacle (see, e.g., Mellon and Parsons (1995),Neuberger (1999)). These papers do not derive equilibrium open interest and futures prices.
4
In what follows, I develop a simple infinite horizon model that captures these ideas
and to explore their empirical content. I present my model in Section I, the definition of
equilibrium in Section II and the solution to the model in Section III. I then discuss the
solution in Section IV. I conclude in Section V.
I The Model
The economy is defined on a continuous time, infinite horizon, T = [0, ∞), with a singlegood which is also used as the numeraire. The underlying uncertainty of the economy
is characterized by a n-dimensional Weiner process wt, t ∈ T . There are two classes ofinvestors denoted by i = 1, 2. Investors are identical within each class but different across
classes in endowments. Let the population weights of these two classes be ω and 1 − ω,respectively, where ω ∈ [0, 1]. Investors in class-i will also be referred to as investor-i.The economy is further specified as follows.
A Investment Opportunities
There are four publicly traded assets in the economy. Investors have access to a risk-free
money-market account that pays a positive, constant rate of return r > 0. In addition,
investors can trade shares of the spot in a competitive spot market, along with two futures
written on the spot in a competitive futures market. The shares of the spot are perfectly
divisible and all assets can be traded at no cost.
Each share of the spot pays a cumulative “convenience yield” Dt.4 Dt is governed by
the process
dDt = Ztdt+ bDdwt, (1)
where Zt = Z1,t + Z2,t and Zj,t (j = 1, 2) follows
dZj,t = −aZjZj,tdt+ bZj
dwt. (2)
Here aZj(j = 1, 2) are non-negative constants and bD and bZj
(j = 1, 2) are matrices of
proper order. We will also call Zt the “fundamental” of the spot as it fully determines the
4I use the term “convenience yield” in the sense of Brennan (1991) and Pindyck (1993)–the valueof any benefits that inventories provide, including the ability to smooth production or facilitate thescheduling of production and sales net storage and other costs. That is, Dt is a “net” convenience yield.The analogy here is that of dividends to holding equity.
5
expectation of future spot payoffs. While our model can be solved for arbitrary values of
aZj, for simplicity, we set aZ1
= 0, so that Z1,t follows a random walk. Then Z1,t is the
permanent component of the fundamental, while Z2,t is the mean-reverting or transitory
component. So when we speak of convenience yield shocks being more transitory, we are
referring to aZ2increasing in magnitude. Let St denote the spot price.
Figure 1 illustrates the time intervals in which the two futures with identical maturity
lengths ofM can be traded. The top line is the maturity cycle for the first contract, while
the bottom line represents the second contract. At m1,k−1, the (k−1)-th replication of thethe first contract has matured. Since the (k−1)-th replication of the second contract isinitiated after the first, it does not mature until m2,k−1. The k-th replication of the first
contract starts trading at t1,k = m1,k−1 and expires at m1,k at which time the (k + 1)-th
contract starts trading at t1,k+1 = m1,k. The k-th replication of the second contract starts
trading at t2,k, a period of length L after the first contract started trading, and matures at
m2,k. This maturity and replication process cycles periodically from the k-th replication
to the (k+1)-th replication and so on.
Our discussion will primarily focus on the solution in the time interval between t = t2,k
and t = m1,k. In this interval, the first contract is nearest-to-maturity. So, for convenience,
we will refer to the first contract as the “nearby” contract and the second contract as
“distant” contract. Let H1,t be the price of the nearby and H2,t be the price of the
distant.
c• c•
m1,k−1
t1,k m1,k
t1,k+1k-th “nearby”
c• c•
m2,k−1
t2,k m2,k
t2,k+1k-th “distant”
Figure 1: Futures Maturity Cycle
B Endowments
Without loss of generality, I assume that each investor is endowed with 1 share of the spot
and the futures are in zero net supply. In addition to these publicly traded securities, it
6
is assumed that investor-1 receives a non-marketed income with a cumulative excess rate
of return qt. qt follows
dqt = Xt"aqXtdt + e"Xtbqdwt, (3)
where Xt = [1, Yt]" and Yt follows
dYt = −aY Ytdt+ bY dwt. (4)
Here aq and aY are positive, bq and bY are matrices of proper order and e = [1, 1]". Yt is
a state variable which leads to time-variation in the expected return and volatility of the
non-marketed income.
For t ≥ 0, let It ≡ {Ds, Z1,s, Z2,s, qs, Ys, Ss, H1,s, H2,s : s ≤ t} denote the fullinformation set about the economy. All investors are endowed with It, that is, they allhave complete and symmetric information sets.
C Preferences
All investors have constant absolute risk aversion (CARA). They maximize the expected
utility of the following form:
i = 1, 2 : Ei,t
!−" ∞
te−ρ(s−t)−γci,s ds
#, (5)
where ρ and γ (both positive) are the time discount coefficient and the relative risk
aversion coefficient, respectively, and ci,s is consumption at time s.
D Distributional Assumptions
I further assume that wt = [wD,t, wZ1,t, wZ2,t, wY ,t, wq,t] and
bD = [σD, 0, 0, 0, 0], bZ1= [0, σZ1
, 0, 0, 0], bZ2= [0, 0, σZ2
, 0, 0],
bY = [0, 0, 0, σY , 0], bq = [σqκDq, 0, 0, 0, σq$1− κ2
Dq].
This specification of underlying shocks has a simple interpretation. For instance, instan-
taneous shocks to the convenience yield Dt are characterized by wD,t and σD gives its
instantaneous volatility. All shocks in the economy are uncorrelated except for conve-
nience yield and investor-1’s non-marketed risks. These two shocks are assumed to be
positive correlated, Corr(dwD,t, dwq,t) = κDq > 0.
7
E Discussion of Key Assumptions
My goal in this paper is to understand the effects of convenience yield fluctuations on
optimal hedging and the term structure of open interest and futures prices. To do so, I
make a number of simplifying assumptions for tractability. A number of these are fairly
standard in the literature. For instance, like many other models of futures markets, we
assume that the interest rate, r, is exogenously specified and assumed to be constant.
The assumption that the spot pays an exogenously specified convenience yield bypasses
the difficulty of dealing with the consumption and production of commodities. This
assumption is in other studies of futures markets such as Brennan (1991), Pindyck (1993),
Gibson and Schwartz (1990) and Schwartz (1997). However, Williams and Wright (1991)
and Routledge, Seppi and Spatt (1996) point out that the convenience yield may also
arise endogenously from a non-negativity constraint on inventory, in which case stockouts
then play an important role in generating state dependent correlation between spot and
futures prices. Although my model does not account for such constraints, my results are
not likely to be colored since the results are about the behavior of the term structure of
open interest and futures prices on average (i.e. on a typical day) while stockouts are
likely to be seasonal effects. Related, the empirical studies which document the stylized
patterns that my model is trying to address are careful to control for seasonal effects.
Importantly, note that I need both σZ1and σZ2
to be greater than zero. That is, I
need the convenience yield fluctuations to be a two-factor model: Z1,t + Z2,t. Otherwise,
trading in the existing securities can span convenience yield fluctuations and markets are
effectively complete. In a sense, if there is not sufficiently rich convenience yield shocks,
investors could do a pretty good job of hedging by trading just the nearby. This seems
consistent with what we see in futures markets such as oil (with rich convenience yield
shocks and lots of trading in a large menu of futures of different maturities) and S&P (with
negligible convenience yield fluctuations and little trading other than in the nearest-to-
maturity futures). If I added more futures of different maturities, I would need to add
additional factors to the convenience yield–an additional factor for every new futures
contract added.
Furthermore, I assume that investors in class-1 are endowed with non-marketed income
qt. Since investors are identical in every other way, this non-marketed risk is a simple way
8
to generate heterogeneity among investors and trade in futures. The important component
of this assumption is that κDq > 0. (The sign of the correlation does not matter as long
as it is not zero.) Because investor-1’s non-marketed income is positively correlated with
spot payoffs, investor-1 prefers to reduce his spot holdings so as to hedge his non-marketed
risk. Investor-2 ends up taking on a larger spot position. As a result, investors then want
to hedge their differing spot positions with futures.5
II Definition of Equilibrium
Given the investment opportunities specified in (1)-(2) and endowments in (3)-(4), in-
vestors choose consumption and investment policies to maximize their expected util-
ity over life-time consumption. Let the consumption policy of investor-i be given by
{ci,t : t ∈ T }. His investment policies in the spot, nearby and distant are given by{θSi,t : t ∈ T }, {θH1
i,t : t ∈ T } and {θH2i,t : t ∈ T } respectively. Furthermore, the excess re-
turn on one share of the spot, which is the return minus the financing cost at the risk-free
rate, follows
dQ0,t = dSt − rStdt+ dDt. (6)
Note that dQ0,t is the excess return on one share of the spot instead of the excess return
on one dollar invested in the spot. The former is the excess share return, while the latter
is the excess rate of return. And the return on the first futures contract is
dQ1,t = dH1,t. (7)
Similarly, the return on the second futures contract is
dQ2,t = dH2,t. (8)
For future convenience, define θi,t to be the vector of investor-i’s investment policies.
That is, θi,t = [θSi,t, θ
H1i,t , θ
H2i,t ]
". Further define Qt to be the vector of returns. That is
Qt = [Q0,t, Q1,t, Q2,t]".
5One can think of the class-1 investors as utility companies that use (spot) natural gas in the productionof electricity. These companies hold inventories of natural gas. One can think of the non-marketed riskas the positions that these utilities have in coal, another input in their production process. Fluctuationsin coal price then affect their spot positions. The companies manage the price risk of these positionswith futures. Think of the class-2 investors (who are not exposed to non-marketed risk) as naturalcounterparties who make the market for the trades of the class-1 investors.
9
Given the investors’ preferences in (5), the return processes defined in (6)-(8), the
investors’ optimization problems are given by: for i = 1, 2
Ji,t ≡ sup{ci,θi}
Ei,t
!−" ∞
te−ρ(s−t)−γci,s ds
#(9a)
subject to dWi,t = (rWi,t − ci,t) dt+ θi,t"dQt + δidqt, (9b)
where Ji,t is investor-i’s value function at time t, Wi,t is investor-i’s wealth and δi is an
index function, δi = 1 if i = 1 and δi = 0 if i = 2.
An equilibrium is given by spot and futures prices such that investors follow their
optimal policies and markets clear:
t ∈ T : ωθ1,t + (1−ω)θ2,t = [1, 0, 0]". (10)
The resulting equilibrium prices and investors’ optimal policies can in general be expressed
as a function of the state of the economy and time. The state of the economy is determined
by the investors’ wealth and their information on current and future investment oppor-
tunities. And due to the assumptions of a constant risk-free rate and constant absolute
risk aversion in preferences, investors’ demand of risky investments will be independent
of their wealth (see, e.g., Merton (1971)). Thus I seek an equilibrium in which the market
prices are independent of investors’ wealth. Let • denote the relevant state variables.Then one can write St = S(•; t), Hj,t = Hj(•; t), and {ci(•; t), θi(•; t)}.Due to the nature of the periodic replication of futures after expiration, I will in this
paper consider periodic equilibria in which the equilibrium price processes and investors’
optimal policies exhibit periodicity in time. (See also Hong and Wang (2000) for an
example of periodic equilibria.) Specifically, since M , the length of a futures, stays the
same across time, one has
Definition 1 In the economy defined above, a periodic equilibrium is defined by the price
functions {S(•; t), Hj(•; t) (j = 1, 2)} and policy functions {ci(•; t), θi(•; t)}, i = 1, 2, suchthat (a) the policies maximize investors’ expected utility, (b) all markets clear, and (c) the
price functions are periodic in time with periodicity M, and (d) investors’ policy functions
are also periodic in time.
Under Definition 1, the k-th replication of a futures contract depends on the underlying
uncertainty in the economy in the same way as the (k + 1)-th replication and so forth:
10
for k = 1, 2, . . ..
S(•; t1,k) = S(•; t1,k+1), Hj(•; tj,k) = Hj(•; tj,k+1). (11)
Realized values of the spot and futures can be different over time as the state variables
change. Furthermore, periodicity in each investor’s optimization problem and policy
functions yield: for i = 1, 2
Ji (•; t1,k) = Ji (•; t1,k+1) . (12)
The periodicity conditions for the prices and value functions, (11) and (12), provide
the necessary boundary conditions we need to solve for a periodic equilibrium. Thus, a
periodic equilibrium is given by periodic price functions (11) such that investors optimally
solve (9), (12), and the markets clear–(10) holds.
Furthermore, I restrict myself to the linear equilibria in which the price functions are
linear in •.
Definition 2 A linear, periodic equilibrium is a periodic equilibrium in which: (a) St =
S(•; t) and Hj,t = Hj(•; t) for j = 1, 2, (b) these price functions are linear in • and (c)time dependent with periodicity given in (11).
Finally, I also impose the following no-arbitrage condition, which states that the fu-
tures price equals the spot price at the expiration date, mj,k, for j = 1, 2 and k = 1, 2, . . .
Hj(•;mj,k) = S(•;mj,k). (13)
This condition provides another set of boundary conditions needed to solve for equilibrium
prices.
III Solution of Equilibrium
In solving for an equilibrium, I proceed as follows: first, conjecture a particular equilib-
rium, then characterize the investors’ optimal policies and the market clearing conditions
under the conjectured equilibrium, and finally verify that the conjectured equilibrium in
fact exists.
11
To begin with, I calculate the expected present discounted value of the convenience
yield to holding the spot:
t ∈ T : F0,t = Et
!" ∞
te−r(s−t)dDsds
#= λ0,Z1
Z1,t + λ0,Z2Z2,t, (14)
where λ0,Z1= 1/r and λ0,Z2
= 1/(r + aZ2). F0,t is simply the spot price in a hypothetical
economy with a risk neutral agent. In this risk neutral economy, I can apply the cost-of-
carry formula to calculate the corresponding futures prices:
t ∈ [tj,k, mj,k] : Fj,t = er(mj,k−t)
%St − Et
!" mj,k
te−r(t−s)dDsds
#&= λj,Z1
(t)Z1,t + λj,Z2(t)Z2,t,
(15)
where λj,Z1(t) = λ0,Z1
and λj,Z2= λ0,Z2
e−aZ2(mj,k−t). Notice that λj,Z1
(t) and λj,Z2(t) are
periodic and converge (as required by no-arbitrage) to the spot price coefficients, λ0,Z1
and λ0,Z2respectively, at t = mj,k
With these calculations in mind, I conjecture that the equilibrium asset prices have
the following linear form:
Conjecture 1 A linear, periodic equilibrium is {St, H1,tH2,t} such that:
St = F0,t − λ0,XXt,
H1,t = F1,t − λ1,XXt,
H2,t = F2,t − λ2,XXt,
(16)
where Xt = [1, Yt]", λ0,X = [λ0,0, λ0,Y ] and for j = 1, 2, λj,X = [λj,0, λj,Y ]. λ0,X and
λj,X (j = 1, 2) are deterministic, time-dependent matrices with appropriate periodicities.
For convenience, define λ(t) to be the following time-varying column vector:
λ = stack{λ0,X", λ1,X
", λ2,X"}, (17)
where the function stack makes a large column vector out of the elements within the
braces.
To characterize the equilibrium, I take the price function in (16) as given and derive
each investor’s conditional expectations, policies and the market clearing conditions.
12
A Optimal Policies
Given the price functions, I can solve for the optimal policies of the investors. Investor-i’s
control problem as defined in (9) can be solved explicitly. The following lemma summarizes
the results.
Lemma 1 Given the price functions in (16), investor-i’s value function has the form:
t ∈ T : Ji,t = exp%−ρt− rγWi,t − 1
2
'Xt
"vi(t)Xt(&, (i = 1, 2), (18)
where vi(t) are symmetric matrices which satisfy a system of ordinary differential equa-
tions given by
vi = gi,v(t; vi(t);λ(t)), (19)
where gi,v is given in the proof in the Appendix. Furthermore, her optimal consumption
policy is
ci,t = −1γlog(r) + rWi,t +
1
2Xt
"vi(t)Xt. (20)
And her optimal investment policy is
θi,t =1
rγhi(t)Xi,t, (21)
where hi is given in the proof in the Appendix.
Given λ(t), the above lemmas expresses investor-i’s optimal policies as functions of the
matrices vi(t), to be solved from (19), which is a (vector) first-order ordinary differential
equation. A periodic solution for investor-i’s control problem further requires that for
k = 0, 1, 2, . . .:
vi(t1,k) = vi(t1,k+1) (22)
B Market Clearing
In equilibrium, the markets must clear. From (10) and Lemma 1, the market clearing
condition requires that
ωh1
rγ+ (1− ω)h2
rγ= stack{[1, 0], [0, 0], [0, 0]}. (23)
13
Given hi (i = 1, 2), (23) defines a (vector) first order differential equation for λ:
λ = gλ(t;λ(t); v1(t), v2(t)), (24)
where gλ is given in the Appendix. The periodicity condition for spot price requires that
λ0,X(t1,k) = λ0,X(t1,k+1). (25)
And the periodicity condition for the futures prices implies that
λj,X(tj,k) = λj,X(tj,k+1). (26)
Additionally, the convergence of futures to spot prices at maturity given in (13) implies
that
λ0,X(mj,k) = λj,X(mj,k). (27)
(27) provides an additional set of boundary conditions that an equilibrium solution must
satisfy.
C Existence and Computation of Equilibrium
The previous discussion characterizes the investors’ optimal policies in a linear, periodic
equilibrium of (16). Solving for such an equilibrium now reduces to solving (19) and
(24), a system of first-order differential equations, for v1, v2 and λ subject to boundary
conditions (22), (25), (26) and (27). The solution of a system of differential equations
depends on the boundary conditions. In the case of the familiar initial-value problem, the
boundary condition is simply the initial value of the system. It seeks a solution given its
value at a fixed point in time. My problem, however, has a different boundary condition. I
need to find particular initial values vi(t1,k) (i = 1, 2) and λ(t1,k) such that the periodicity
conditions hold. This is known as a two-point boundary value problem, which seeks a
solution of the system with its values at two given points in time satisfying a particular
condition.
Theorem 1 states the result on the existence of a solution to the given system, which
gives a linear, periodic equilibrium of the economy. Here, the condition that ω be close
to one arises from the particular approach I use in the proof as opposed to economic
rationales (see Appendix). My proof relies on a continuity argument. It is first shown
14
that at ω = 1, a solution to the given system exists. Since the system is smooth with
respect to ω, it is then shown that a solution also exists for ω close to one. I do not specify
in the proof, however, how close it has to be.
Theorem 1 For ω close to one, a linear periodic equilibrium of the form in (16) exists
generically in which the optimal policies of both investors are given by Lemma 1.
In general, the model needs to be solved numerically. The numerical method used to solve
this system of nonlinear first order differential equations is standard and is discussed in
Kubicek (1983).
IV The Term Structure of Open Interest and Futures
Prices
In this section, I develop the implications of the model for the term structure of open
interest and futures prices. I begin in Section IV.A below by setting the volatility of the
persistent component of non-marketed income shock Yt equal to zero, i.e. σY = 0. So
the non-marketed income process is simply dqt = σqdwq,t, which is i.i.d. over time. This
setting is interesting for a couple of reasons. First, I can obtain closed-form solutions for
prices and holdings. These closed form solutions will help us greatly in interpreting our
numerical solution for σY > 0. It turns out that a number of our key results derived for
σY = 0 are qualitatively similar for non-zero values of σY . When σY is very large, we
obtain a few new predictions. We present the numerical solution for the case σY > 0 in
Section IV.B below. In the discussions of these two cases, we relate our results to the
empirically documented variation in the term structure of open interest and futures prices
across futures markets.
A I.I.D. Non-marketed Income Shocks: σY = 0
I begin with the the following proposition regarding equilibrium prices.
Proposition 1 When non-marketed risk shocks in the economy are i.i.d, the spot price
St is given by
St = F0,t − λ0,0 (28)
15
where λ0,0 is given by
λ0,0 = γ(λ20,Z1σ2
Z1+ λ2
0,Z2σ2
Z2+ σ2
D) + γωσDq. (29)
The futures prices Hj,t (j = 1, 2) are given by, for k = 1, 2, 3, ...,
t ∈ [tj,k, mj,k] : Hj,t = Fj,t − λj,0(t), (30)
where λj,0(t) is given by
λj,0(t) = λ0,0 + γ'(mj,k − t)λ2
0,Z1σ2
Z1+1− e−aZ2
(mj,k−t)
aZ2
λ20,Z2σ2
Z2
(. (31)
Notice that the solution for St trivially satisfies the definition of a linear periodic equi-
librium since λ0,0 is constant through time. Moreover, the solution for Hj,t also satisfies
the periodicity condition. As required for no-arbitrage, the futures price converges to the
spot price at expiration: Fj,t = F0,t at expiration and λj,0(t) = λ0,0 at t = mj,k.
There is a simple economic interpretation for the spot and futures prices. Recall
from the discussion in Section III of equation (14) that the first piece of the spot price,
F0,t, is simply the expected present discounted value of the payoffs to holding the spot.
The second piece, λ0,0, is the price discount, which naturally increases with investor risk
aversion γ and the variance of convenience yield shocks (σD, σZ1and σZ2
).
Moreover, the price discount also increases with σDq, the covariance of the spot pay-
off with investor-1’s non-marketed income. Because of the positive correlation between
investor-1’s non-marketed income shocks and convenience yield shocks, investor-1 wants
to reduce his initial holdings of the spot for diversification reasons. Given a fixed supply
of the spot and given that the demand of investor-2 has not changed, for markets to clear,
the spot price has to drop (or the expected return to holding the spot increase) to induce
investor-2 to hold more shares of the spot in equilibrium.
We next turn to the futures prices Hj,t (j = 1, 2). From the discussion in Section III
of equation (15), the first piece, Fj,t, is simply the futures price in a risk neutral economy.
The second piece, λj,0(t), is the price discount for being long the futures. It increases with
investors’ risk aversion and volatility of convenience yield shocks. In other words, there is
a positive futures risk premium to being long futures, just as in other models with normal
backwardation along the lines of Keynes and Hicks.
16
We next consider the holdings of the investors at these prices. Without loss of gener-
ality, we will focus our analysis on the holdings of investor-1. The following proposition
establishes the equilibrium holding patterns.
Proposition 2 Given the equilibrium prices specified in Proposition 1, we can establish
the following results regarding the equilibrium holdings. The spot position of investor-1 is
given by
θS
1,t = 1− (1− ω)σqκDq
σD
. (32)
And investor-1 takes a long position in the nearby futures θH11,t > 0 and simultaneously a
short position in the distant futures θH21,t < 0.
Both investors-1 and 2 are endowed with one share of the spot. Because investor-1’s
non-marketed income is positively correlated with payoffs to the spot, investor-1 wants
to reduce his initial spot holdings (away from one) to hedge his non-marketed income.
This reduction is proportional to the regression coefficient of the instantaneous return to
the non-marketed income dqt on the convenience yield: (σqκDq)/σD. The larger is this
coefficient, the more aggressively investor-1 hedges his non-marketed income risk using
his spot holdings. In equilibrium, investor-1 holds less than one share and investor-2 ends
up holding more than one share, by an amount equal to
(1− ω)σqκDq
σD
.
Note that if ω = 1 (all investors are from class-1), then no hedging is possible.
As a result, investors-1 and 2 will have different demands for the futures. In particular,
investor-1’s spot position is subject to spot price risk in the form of convenience yield
fluctuations driven by Z1,t and Z2,t. As a result, investor-1 hedges these risks with futures
contracts. The natural thing for investor-1 to do is to hedge his (short) spot position with
a long position in the nearby futures. However, when the convenience yield growth rate
follows a two factor model, positions in the nearby is subject to a basis risk as well. So,
the distant contract provides a vehicle to hedge this basis risk. Since she goes long in the
nearby contract, the natural way to hedge this long position in the nearby is with a short
position in the distant contract.
17
Having established the equilibrium prices and holdings in Propositions 1 and 2, we now
derive a number of results regarding the term structure of open interest. We calculate
the open interest in each futures by taking the absolute value of investor-1’s position
and weighing it by the proportion of class-1 investors in the economy, ω. The following
proposition summarizes the main results on open interest.
Proposition 3 When non-marketed income shocks are i.i.d., the open interest of the
nearby futures is greater than that of the distant futures. And the ratio of the open interest
in the distant to the nearby futures is increasing in the mean reversion of convenience yield
shocks.
These results reflect the fact that when shocks to Z2,t are more persistent (or aZ2→ 0),
spot and futures prices are more correlated and hence the nearby contract is more effective
in hedging both convenience yield shocks, Z1,t and Z2,t. As aZ2increases, the nearby
contract is a less effective hedge and there is more need to use the distant contract to
hedge the basis risk.
Because prices are all determined endogenously, our model also have a number of
implications for the term structure of futures prices.
Proposition 4 When non-marketed income shocks are i.i.d., the return volatility of the
nearby futures is greater than that of the distant futures, i.e. the Samuelon effect holds.
And the ratio of the return volatility of the nearby to the distant futures is increasing in
the mean reversion of convenience yield shocks.
When the non-marketed risks are i.i.d., the only uncertainty affecting the spot and futures
prices are Z1,t and Z2,t. When the convenience yield is mean reverting, aZ2> 0, the
price elasticity to Z2,t of the nearby futures is greater than that of the distant futures:
λ1,Z2> λ2,Z2
. The intuition is simple. Since shocks to Z2,t are transitory, contracts far
from maturity will be less sensitive to these shocks as they will die out by the time the
contract expires. Hence, the return volatility of the distant futures is less than that of
the nearby, or the Samuelson Effect holds (see, e.g., Samuelson (1965)). An immediate
implication of this analysis is that the more transitory are convenience yield shocks, the
greater the ratio of the return volatility of the nearby to the distant futures.
18
In addition to these implications regarding the term structure of futures price volatility,
our model also has a number of implications regarding the futures risk premium.
Proposition 5 When non-marketed income shocks are i.i.d., the futures risk premium is
increasing in the volatility and persistence of convenience yield shocks.
It is easy to see this result from the closed-form solution for λj,0 given in Proposition 1.
Since λj,0 > 0, it follows that long positions earn a positive expected (excess) return (i.e,
a futures risk premium). The greater is λj,0, the greater the futures risk premium. It is
easy to see that λj,0 increases with σZj(j = 1, 2), the volatility of the convenience yield
shocks. It is not hard to show that λj,0 also increases with aZ2.
In passing, it is worth noting that a number of the predictions of Propositions 2-5 fit
nicely with existing evidence on hedging strategies, open interest and futures prices. To
begin with, the hedging strategy described in Proposition 2 of being on opposite sides of
nearby and distant futures (long nearby and short distant or vice versa) is known as a
spread trade (see, e.g., Brown and Errera (1987) for a description of this strategy). Our
model suggests that this hedging strategy is most likely to be used in futures markets with
a substantial convenience yield and hence basis risk to rolling nearby contracts. Indeed,
consistent with the model, such a strategy is most often used in energy futures, markets
notorious for convenience yield fluctuations and basis risk associated with rolling over
nearby contracts.
And consistent with the predictions of Proposition 3, oil markets, which have very
mean reverting convenience yield shocks, have open interest evenly distributed among
nearby and distant. In contrast, metals and agriculturals, which tend to have more
persistent shocks, have more of their open interest concentrated in the nearby contracts.
It is important to note that the implications of Propositions 2-3 are not entirely con-
sistent with empirical findings. In many markets, when the nearby contract is within a
month or so of expiration, traders roll out of their positions in the nearby into the next
closest contract. Many argue that this effect is likely due to liquidity considerations. In-
deed, contracts within the last month of expiration tend to be very illiquid. My model
does not capture this effect. That is, open interest in the nearby exceeds the distant fu-
tures until expiration at t = mj,k, at which point the distant futures becomes the nearby
19
and a new contract is added. Hence, one should really think of the predictions of Propo-
sitions 2-3 as pertaining to the relationship between contracts of different maturities at a
single point in time that is far from the expiration of the nearby.
Consistent with some of the predictions of Proposition 4, there is a more pronounced
Samuelson effect in futures markets with more mean reverting convenience yield shocks.
For instance, the Samuelson effect is more pronounced in energy futures, less so in agri-
culturals, even less so in metals and almost non-existent in financial futures (see, e.g.,
Bessembinder, et.al. (1996)). There is less anecdotal evidence in support of Proposition
5. Its implications, however, are testable.
B Persistent Non-Marketed Income Shocks: σY > 0
We now discuss the solution when σY > 0. The equilibrium spot price is
St = F0,t − λ0,0 − λ0,Y1Yt, (33)
where λ0,0 > 0 and λ0,Y1> 0 are periodic in time. The price discount now consists of
two parts: (a) an unconditional part, λ0,0, and (b) a conditional part that depends on the
state variable driving non-marketed risk. The conditional covariance between convenience
yield shocks and non-marketed income shocks becomes (1 + Yt)σDq. Yt is a state variable
which drives the amount of selling or buying that the investor-1 does since Yt drives the
sign and the magnitude of the correlation between investor-1’s non-marketed income and
spot payoffs. Again, the price of the spot has to drop to attract class-2 investors to take
off-setting positions. Note that this price changes occurs without any change in the spot’s
payoffs. Therefore, the equilibrium spot price depends not only on the convenience yield
but also on class-1 investors’ non-marketed risk Yt.
The equilibrium futures prices are now given by (for j = 1, 2):
Hj,t = Fj,t − λj,0 − λj,Y1Yt, (34)
where λj,0 > 0 and λj,Y1> 0 are periodic in time. The dependence of the futures prices
on Yt arises from the fact that investors trade in the futures to help allocate the risks
associated their different spot positions.
Unfortunately, we are no longer able to obtain closed-form solutions. In general, the
price coefficients need to be solved numerically. To discuss some of the properties of
20
the model, I set the parameters of the model as follows for the numerical exercise. The
parameter of absolute risk aversion, γ, is set to be 100. I choose ω = 0.5 so that the
number of class-1 traders is one-half of the population. The maturity length for a given
contract,M , is set to be 0.6. The second contract will be staggered by a length of L = 0.1.
I will choose the other parameters such that M = 0.6 corresponds roughly to a contract
with seven months to expiration. I set the constant risk-free rate, r, to be .05. I set the
mean reversion coefficient for the convenience yield, aZ2, to be 1 and the mean reversion
coefficient for the non-marketed risk shocks, aY , to be 3. We set σY = σZ1= σZ2
= 0.05,
σD = σq = 0.25 and κDq = 0.5.
Figure 2 illustrates the price coefficients, solved for numerically at these parameters.
Figure 2(a) shows the solutions for λ0,0, λ1,0 and λ2,0. The solution for λ0,0 is given by
the solid line. The solutions for λ1,0 (the nearby) and λ2,0 (the distant) are given by
dashed and ’+’ respectively. One the x-axis is the interval [0, .5]. Moving from t = 0 to
t = .5 is the same as moving from t = t2,k to t = m1,k (see Figure 1). Notice that at the
expiration of the nearby, λ1,0 equals λ0,0. Moreover, λ2,0 > λ1,0 > λ0,0 as in the case of
i.i.d. non-marketed income shocks.
Figure 2(b) shows the solutions for λ0,Y , λ1,Y and λ2,Y , which are represented again by
the solid, dashed and ’+’ lines, respectively. At the expiration of the nearby, λ1,Y = λ0,Y .
Before the expiration date, the distant futures is less sensitive to Yt shocks than the nearby
futures: λ2,Y < λ1,Y < λ0,Y . One reason for this is the same as why the distant futures
is less sensitive to mean reverting convenience yield shocks. The distant futures is less
sensitive to mean reverting non-marketed income shocks than the nearby because these
shocks are more likely to die away by the time the distant futures expires.
Next, I focus on the investors’ optimal policies. First, the investors’ optimal investment
policies in the spot are given by
θS
i,t = hS
i,0 + hS
i,Y Yt. (35)
The first component hSi,0 gives his unconditional spot position. The second compo-
nent, hSi,1Yt, arises from hedging (market-making) trades associated with investor-1’s non-
marketed risk. The investors’ optimal policies in nearby and distant futures are given
21
λ0,0,λ1,0,λ2,0 λ0,Y ,λ1,Y ,λ2,Y
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5108
108.5
109
109.5
110
110.5
111
111.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.005
0.01
0.015
0.02
0.025
0.03
(a) (b)
Figure 2: Price Coefficients. In figure (a) the spot and futures price risk discounts are
shown: λ0,0 is the solid line, λ1,0 is the dashed line, λ2,0 is the ’+’ line. In figure (b) the spot
and futures price elasticities to Yt are shown: λ0,Y is the solid line, λ1,Y is the dashed line,
λ2,Y is the + line. On the x-axis is the time interval [0, .5] which corresponds to [t2,k, m2,k]
(see Figure 1). The remaining parameters are set at the following values: r = 0.05, aZ2= 1,
aY = 3, σZ1= 0.05, σZ2
= 0.05, σY = 0.05, σD = 0.25, σq = 0.25, κDq = 0.5, M = 0.6,
L = 0.1, ω = 0.5, γ = 100.
byθH1i,t = h
H1i,0 + h
H1i,YYt,
θH2i,t = h
H2i,0 + h
H2i,YYt.
(36)
Since Yt is on average zero, I will be particularly interested in the unconditional portion
of the investors’ holdings hSi,0, h
H1i,0 and h
H2i,0 to contrast with those derived for the case of
σY = 0.
Figure 3 shows these unconditional positions in spot, the nearby and the distant futures
respectively. Figures 3(a) shows the solutions for hS1,0 and h
H11,0. Figure 3(b) shows the
solution for hH21,0. Notice that h
S1,0 is less than 1, h
H11,0 > 0 and h
H21,0 < 0. Qualitatively, these
patterns are quite similar to those described in Proposition 2. Investor-1 holds less than
one share of the spot, longs the nearby futures and shorts the distant futures. Moreover,
|hH11,1| > |hH2
1,2| as in Proposition 3. So, when σY is not very large, the qualitative patterns
in Proposition 3 regarding open interest continue to hold.
When σY is very large, this need no longer be the case. Investor-1’s hedging problem
becomes considerably more complex. In this instance, investor-1 may actually find it
optimal to take a long position in the distant futures to hedge his spot position as opposed
22
hS1,0, h
H11,0 hH2
1,0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
(a) (b)
Figure 3: Spot and Futures Holdings. In figure (a) the spot and nearby futures holdings
are shown: hS1,0 is the solid line, h
H11,0 is the dashed line. In figure (b) the distant futures
holding is shown: hH21,0 is the ’+’ line. On the x-axis is the time interval [0, .5) which
corresponds to [t2,k,m2,k), the interval in which contract 1 is the nearby and contract 2 the
distant (see Figure 1). The remaining parameters are the same as in Figure 2.
to taking a long position in the nearby. We consider the solution for σY = 0.8 in Figure
4. All the other parameters remain the same. Figure 4(a) shows the solutions for hS1,0
and hH11,0 and Figure 4(b) shows the solution for h
H21,0. Notice that h
S1,0 is still less than 1,
but hH11,0 < 0 and h
H21,0 > 0. The other interesting to note is that the absolute magnitude
of the average position in the distant futures is now greater than in the nearby. As such,
it turns out that the distant futures can actually attract more open interest than the
nearby. Hence, in futures markets with substantial non-marketed risks (σY large), open
interest tends to be evenly distributed between nearby and distant futures. Indeed, the
open interest in distant futures can even exceed that in the nearby.
Finally, when non-marketed income shocks are persistent, futures price volatility de-
pends not only on the convenience yield shocks Zi,t (i = 1, 2) but also on Yt. The
Samuelson Effect still characterizes the term structure of futures price volatility. And the
more mean reverting the convenience yield and non-marketed income shocks, the more
prominent the Samuelson effect. The futures risk premium now also depends the σY in
addition to the volatility and persistence of convenience yield shocks. Not surprisingly,
numerical comparative statics indicate that the results of Proposition 5 continue to hold
when σY > 0.
23
hS1,0, h
H11,0 hH2
1,0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
−1.5
−1
−0.5
0
0.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(a) (b)
Figure 4: Spot and Futures Holdings for σY Large. In figure (a) the spot and nearby
futures holdings are shown: hS1,0 is the solid line, h
H11,0 is the dashed line. In figure (b) the
distant futures holding is shown: hH21,0 is the ’+’ line. On the X-axis is the time interval
[0, .5) which corresponds to [t2,k, m2,k), the interval in which contract 1 is the nearby and
contract 2 the distant. The remaining parameters are the same as in Figure 2 except that
σY = 0.8.
V Conclusion
This paper develops a model of a futures market to study the effects of convenience
yield shocks on optimal hedging and the resulting term structure of open interest and
futures prices. Because convenience yield fluctuations generate basis risk to trading fu-
tures, investors, continuously facing spot price risk over time, have to simultaneously
trade futures of different maturities to optimally hedge their spot positions. The model
generates a number of predictions that are consistent with empirical findings on how the
term structure of open interest and futures prices vary across futures markets.
An important avenue to consider in future research is to what extent liquidity effects
play a role in shaping the term structure of open interest and futures prices. While
our model can qualitatively deliver on different open interest and price patterns, one
may need to account for the multiplier effects associated with liquidity to match these
patterns quantitatively. For instance, if there is asymmetric information or some fixed
cost of participation in particular futures, then there might be bunching of open interest
in a particular contract (see, e.g., Admati and Pfleiderer (1988), Pagano (1989)). I leave
this for future research.
24
Appendix
We begin by introducing some additional notation. Given a matrix m, let tr(m) be its
trace, [m] the column matrix consisting of its independent elements and )m) = max |mi,j|its norm. Whenm is positive semi-definite (positive definite), we statem ≥ 0 (> 0). Also,let i
(m,n)ij be an index matrix of order m × n with its (ij)-th element being 1 and all the
other elements being zero. And let i(n) be an identity matrix of rank n. Θ = {r > 0, ρ >0, γ > 0, aZ1
≥ 0, aZ2≥ 0, aY ≥ 0, σD ≥ 0, σZ1
≥ 0, σZ2≥ 0, σY ≥ 0, σq ≥ 0, 0 < κDq <
1,M > 0, L > 0} denotes the set of parameter values and θ ∈ Θ denotes an element.
A Mathematical Preliminaries
In deriving several results in the paper, we often encounter the two-point boundary-value
problem for a (vector) first-order ODE. Here, we give a formal and relatively general defi-
nition of the two-point boundary-value problem and state some known results concerning
its solution.
Definition A.1 Let f : *+ ⊗ *n ⊗ *m ⊗ * → *n, and g : *n ⊗ *m ⊗ * → *n. Atwo-point boundary-value problem is defined as z = f(t, z; θ,ω) ∀ t ∈ [0, T ]
0 = g [z(0), z(T ); θ,ω] ,(A.1)
where T > 0, θ ∈ Θ and ω ∈ [0, 1].
We also define the terminal value problem: z = f(t, z; θ,ω) ∀ t ∈ [0, T ]z(T ) = zT .
(A.2)
Under appropriate smoothness conditions on f(t, z; θ,ω), (A.2) has a unique solution
z = z(t; θ,ω; zT ), which is differentiable in zT (see Keller (1992), Theorem 1.1.1). Solving
the two-point boundary-value (A.1) is to seek a value for zT that solves
0 = g [z(0; θ,ω; zT ), zT ; θ,ω] ≡ g ◦ z(zT ; θ,ω). (A.3)
The existence of a root to (A.3) relies on the properties of g ◦ z(zT ; θ,ω). Furthermore,let (g ◦ z + 1)(zT ; θ,ω) ≡ g ◦ z(zT ; θ,ω) + zT .
25
Lemma A.1 If (g ◦ z + 1)(·; θ,ω) : *n → *n is continuous and there exists a nonempty,closed, bounded, and convex subset of *n, L, such that (g◦z+1)(·; θ,ω) maps L into itself,then (A.3) has a root and the two-point boundary value problem (A.1) has a solution.
Proof. Existence of a root to (A.3) follows from Brouwer’s Fixed Point Theorem (see,
e.g., Cronin (1994, p.352)).
The condition on (g ◦ z+1) required by Lemma A.1 is not always easy to verify, in whichcase the existence of a solution to (A.1) is not readily confirmed. However, if a solution
exists for ω0, the existence of a solution for ω close to ω0 is easy to establish.
Definition A.2 z = z(t; θ;ω0) is an isolated solution of system (A.1) if the linearized
system y = ∇z f(t, z; θ,ω0) ∀ t ∈ [0, T ]0 = ∇z g(z0; θ,ω0) y(0) +∇z g(zT ; θ,ω0) y(T )
(A.4)
has y = 0 as the only solution, where ∇ denotes the partial derivative operator.
Lemma A.2 Suppose that (a) (A.1) has an isolated solution z = z(t; θ,ω0) for ω = ω0,
and (b) f(t, ·; θ,ω) and g(·; θ,ω) are continuously differentiable in the neighborhood of(t, z(t; θ,ω0),ω0). Then, (A.1) has a solution for ω close to ω0.
Proof. See Keller (1992), p.199.
For future use, we also state two auxiliary lemmas, which are needed in proving Lemma
1 and Theorem 1.
Definition A.3 Let a0 ≥ 0 and a2 > 0 be constant, symmetric matrices. Also let u be
the variable of interest, which is a symmetric matrix. Finally, let a3(t, u) be a positive,
linear operator mapping symmetric matrices into themselves. A matrix Riccati differential
equation is defined as
u+ a0 + (a"1u+ua1)− ua2u+ a3(t, u) = 0 (a.e.) ∀ t ∈ [0, T ] (A.5)
where u = dudtand u(T ) = uT .
26
Lemma A.3 For any given terminal value uT ≥ 0, the matrix Ricatti equation (A.5)
has a unique, symmetric, positive, semi-definite solution. Let m be an arbitrary (bounded
measurable) matrix defined on [0, T ] and k(t) be the solution of the following linear equa-
tion:
k + a0 + (a1−m)"k + k(a1−m) +m"a2m+ a3(t, k) = 0 ∀ t ∈ [0, T ] (A.6)
where k(T ) = uT . If u(t) is the solution of (A.5), then u(t) ≤ k(t) ∀ t ∈ [0, T ].
Proof. See Wonham (1968).
B Solution of Investors’ Control Problem
Let aX = diag{0, aY } and bX = stack{0, bY }. Then
dXt = −aXXtdt+ bXdwt. (A.7)
Given the price function in (16), the excess share return of the spot is
dQ0,t = a0,QXtdt+ b0,Qdwt, (A.8)
where a0,Q = λ0,X
'ri(2) − aX
(− λ0,X and b0,Q = bD + λ0,Z1
bZ1+ λ0,Z2
bZ2− λ0,XbX.
The return on futures is
dQj,t = aj,QXtdt+ bj,Qdwt (A.9)
where aj,Q = −λj,XaX − λj,X and bj,Q = λj,Z1bZ1+ λj,Z2
bZ2− λj,XbX.
Then we re-write Qt = [Q0,t, Q1,t, Q2,t] as
dQt = aQXtdt+ bQdwt (A.10)
where aQ = stack{a0,Q, a1,Q, a2,Q} and bQ = stack{b0,Q, b1,Q, b2,Q}.Investor-i’s control problem can be solved explicitly. We start by conjecturing that
his value function has the following form:
Ji,t = − exp%−ρt− rγWi,t − 1
2Xi,t
"viXt&
(A.11)
27
where vi are symmetric matrices. We show that the conjectured value function gives the
solution to investor-i’s control problem by verifying that it satisfies the Bellman equation:
t ∈ T : 0 = supci,θi
,−e−ρt−γci + Ei,t [dJi,t] /dt
-s.t. dWi,t = (rWi,t−ci,t)dt+ θi,t"dQt + δidqi,t
(A.12)
and the required boundary conditions.
Substituting the conjectured form of the value function into the Bellman’s equation
and applying Ito’s lemma, we obtain the following expression for the optimal policies.
The consumption policy is given by
ci,t = −1γlog(r) + rWi,t +
1
2γXt
"vi(t)Xt. (A.13)
And the investment policies are given by
θi,t =1
rγhi(t)Xt (A.14)
where
hi(t) = (bQbQ")−1 [aQ − bQ(rγebqδi + vibX)
"] .
Substituting the optimal policies into the Bellman’s equation, we obtain equality with the
conjectured form gotten when vi is defined:
0 = vi − (rγebqδi + vibX)(rγebqδi + vibX)" + hi"bQbQ
"hi
− rvi − (aXvi + viaX") + 2aqδi + [v + tr(bXbX
"vi)]i(2,2)11 ;
(A.15)
where v = 2(ρ + r log(r) − r). gi,v of equation (19) is given by (A.15). The periodicboundary condition is of course given by (22).
We now prove the existence of a periodic solution vi for system (19) with boundary
conditions (22) assuming that λ is periodic.
Lemma B.1 There exist αi and βi with αi < 1 such that given the terminal value vi,t1,k+1,
which is symmetric and semi-positive definite, (A.15) has a solution vi(t; vi,t1,k+1) which is
also symmetric, positive semi-definite and )vi(t; vi,t1,k+1)) ≤ αi)vi,t1,k+1
) + βi. Moreover,the system (19) with boundary conditions (22) has a solution.
28
Proof. For simplicity, we will assume that aq > (rγ)2ebqbq
"e". Then using the notation of
Lemma A.3, let u = vi,
a0 = (aQ − rγbQbq"e"δi)"(bQbQ
")−1(aQ − rγbQbq"e"δi) + 2aqδi − (rγ)2ebqbq "e"δi,
a1 = −12ri(2) − aX − (aQ − rγbQbq
"e"δi)"(bQbQ")−1bQbX − rγebqbX
"δi,
a2 = bXbX" − (bQbX
")(bQbQ")−1(bQbX
"),
and a3(t, u) = 2aq + [v + tr(bXbX"u)] i(2,2)
11 . It is easy to verify that a0 and a2 and positive
definite. Assume v ≥ 0 (this lemma can easily be shown to hold for v < 0). a3(t, x) is a
linear positive operator since trace is a linear operator. Hence, (A.15) is a matrix Ricatti
differential equation.
By Lemma A.3, given vi(t1,k+1) = vi,t1,k+1> 0, vi(t; vi,t1,k+1
) exists and is symmetric,
positive semi-definite. Let m = −'(aQ − rγbQbq
"e"δi)"(bQbQ")−1bQbX − rγebqbX
"δi(, then by
Lemma A.3, vi(t; vi,t1,k+1) ≤ k(t; vi,t1,k+1
), where k is the solution to a linear system given
in Lemma A.3 with a1 −m = −12ri(2) − aX < 0.
Letting T = t1,k+1−t1,k, by standard linear differential equation theory, it follows thenthat )k(t; vi,t1,k+1
)) ≤ αi)vi,t1,k+1)+ βi, where
αi = exp{−(r+2aY )T}; ζi = exp{[−12ri(2) − aX](T−s)};
βi =
....." T
0ζi[m(T−s)"a−1
2 m(T−s) + a3(T−s) + a0(T−s)]ζids......
Given the terminal boundary condition vi,t1,k+1, vi(t, vi,t1,k+1
) defines a mapping of Ai :
Li → Li where Li is a non-empty, closed, bounded convex subset of a finite dimensional
normed vector space. It is easy to show that ||Ai(vi,t1,k+1)|| ≤ βi/(1 − αi). By Lemma
A.1, it follows that system (19) with boundary condition (22) has a solution.
A solution to system (19) with boundary condition (22) gives a solution to the Bellman
equation under the desired boundary condition. This completes our proof of Lemma 1.
C Proof of Theorem 1
I prove Theorem 1 as follows. First, I show that there exists a linear periodic equilibrium
at ω = 1. At ω = 1, the value function of investor-1 is given by the following algebraic
29
equation:
0 = −(rγebq + v1bX)(rγebq + v1bX)" + stack{[b0,Qb0,Q", 0], [0, 0]}
− rv1 − (aXv1 + v1aX") + 2aq + [v + tr(bXbX
"v1)]i(2,2)11 ;
(A.16)
Given λ0,X, the equation for v1 has only two roots, one positive and one negative. The
positive root corresponds to the optimal solution of the investors’ control problem since
it gives a higher expected utility. λ can then be solved from
h1
rγ= stack{[1, 0], [0, 0], [0, 0]}.
Given the solution for λ, then v2 can be obtained as the solution to (A.15) and (22), which
is guaranteed by Lemma 1. Hence there is a solution for v1, v2 and λ at ω = 1.
For i = 1, 2, let zi = [vi] for t ∈ T . Let z = stack{z1, z2, λ}. Using the notation in De-finition A.1, (19) and (24) define f . (22) and (25)-(27) define g. Then z is the solution to
a two-point boundary value problem as defined in Definition A.1. Let ω0 = 1. Existence
of z(t; θ,ω0) was just proved above. It remains to verify that z(t; θ,ω0) is an isolated so-
lution. This is equivalent to showing that m(θ,ω0) = ∇zg(z0)+∇zg(zT ) exp{/ t1,k+1
t1,k∇zf}
is nonsingular [see Keller (1992), p.191]. It is easy to show that det(m(θ0,ω0)) /= 0 for
various values of θ0. By Lemma A.2, Theorem 1 holds.
D Proof of Propositions 1-5
Proof of Proposition 1
Take the solutions for St, H1,t and H2,t given in Proposition 1. One then calculates a0,Q
and b0,Q specified in (A.8) and aj,Q and bj,Q specified in (A.9). With these calculations,
calculate aQ and bQ specified in (A.10). With aQ and bQ, we then calculate θi,t given in
(A.14). Since σY = 0, it follows that
θi,t =1
rγ(bQbQ
")−1[aQ − bQ(rγebqδi)"]. (A.17)
Then verify that
ωθ1,t + (1− ω)θ2,t = stack{[1, 0], [0, 0], [0, 0]}.It follows that the solutions for St, H1,t and H2,t given in Proposition 1 are indeed equi-
librium prices.
30
Proof of Proposition 2
In proving Proposition 1, we had to calculate θi,t given in (A.17). Lets explicitly write
down this calculation for investor-1 to prove Proposition 2. For simplicity, lets define
T = m2,k+1 − t and T ∗ = m1,k+1 − t. Note that T , the time-to-maturity of the distantcontract is greater than T ∗, the time-to-maturity of the nearby contract: T > T ∗.
The closed-form solution for θ1,t = stack{θS1,t, θ
H11,t , θ
H21,t} is given by the following. The
spot holding is given by
θS
1,t = 1− (1− ω)σqκDqσD
. (A.18)
And the holdings in the nearby (θH11,t) and the distant (θ
H21,t) are given by
θH11,t =
σqκDqσD
1− e−aZ2T
e−aZ2T ∗ − e−aZ2
T
θH21,t = −
σqκDqσD
1− e−aZ2T ∗
e−aZ2T∗ − e−aZ2
T
(A.19)
From these closed-form expressions, it is easy to verify that θH11,t > 0 and θ
H21,t < 0.
Proof of Proposition 3
Using the expressions for θH11,t and θ
H21,t given in (A.19), the open interest in each futures is
merely the absolute value of these expressions weighted by ω. Since θH11,t > 0 and θ
H21,t < 0,
the ratio of the open interest in the distant over the nearby is given by:
−θH21,t
θH11,t
=1− e−aZ2
T ∗
1− e−aZ2T< 1. (A.20)
Hence the open interest in the distant is less than in the nearby.
Now, take the derivative of this ratio of distant to nearby with respect to aZ2, which
gives us
eaZ2(−T ∗+T )(T (1− eaZ2
T ∗) + T ∗(−1 + eaZ2T ))
(−1 + eaZ2T )2
(A.21)
It is not hard to verify that the expression
T (1− eaZ2T ∗) + T ∗(−1 + eaZ2
T ) (A.22)
is positive. To see this, at aZ2= 0, the expression is zero. Moreover, the derivative of this
expression with respect to aZ2is given by
−eaZ2T ∗T ∗T + eaZ2
TT ∗T > 0. (A.23)
31
Hence, we conclude that as aZ2increases, so does the ratio of open interest in the distant
to the nearby.
Proof of Proposition 4
The ratio of the instantaneous return volatility of the nearby to distant is
b1,Qb1,Q"
b2,Qb2,Q"=λ2
0,Z1σ2
Z1+ λ2
1,Z2σ2
Z2
λ20,Z1σ2
Z1+ λ2
2,Z2σ2
Z2
(A.24)
The price elasticity of the futures with respect to Z2,t is λj,Z2= e−aZ2
(mj,k−t)λ0,Z2for
j = 1, 2. Since m1,k − t < m2,k − t, it follows that λ1,Z2> λ2,Z2
and so the price volatility
of the nearby is greater than that of the distant. Taking the derivative of the ratio with
respect to aZ2and with some tedious algebra, one can show that this derivative is positive,
so the ratio of return volatility in the nearby to distant increases with aZ2.
Proof of Proposition 5
From Proposition 1, take the derivative of λj,0(t) with respect to σ2Zj(j = 0, 1) and it is
clear that the derivative is positive. So, λj,0 increases with σ2Zj. Take the derivative again,
but this time, with respect to aZ2. It is easy to verify that the smaller the aZ2
, the higher
is λj,0.
32
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